The Hebrew calendar is constructed from the simple multiple of a constant
value for the mean lunation period. The mean lunation period is an
averaging of the time taken by the moon to cycle from lunar conjunction to
lunar conjunction. The value used for the constant is 29d 12h 793p.

Hebrew years can be either 12 or 13 lunation periods.

Years of 12 lunation periods are usually called common years.
Years of 13 lunation periods are usually called leap years.

Then the value of L(i) can be either 354d 8h 876p, if year
i is common,
or 383d 21h 589p if year i is leap.

Since the common and leap years are arranged in a 19 year cycle,
L(i) = L(i + 19*k) for any value of i and k.

The Leap Year Distribution

The fixed Hebrew calendar has one, and only one, order for the
arrangement of its common and leap years. The apparent order in which the
leap and common years occur in a given cycle of 19 consecutive years
depends entirely on which element of the arrangement pattern is selected
as first.

Allowing L to represent a leap year and c to represent
a common year, then the arrangement of leap years and common years,
extracted from some starting year in a Hebrew year time line can be seen
to be

...ccLccLcLccLccLccLcL ccLccLcLccLccLccLcL ccLccLcLccLccLccLcL c...

If the pattern had started with the 4th year of the above time line then the arrangement from the first year of the pattern would appear to be

....ccLcLccLccLccLcLccL ccLcLccLccLccLcLccL ccLcLccLccLccLcLccL c...

Looking at the 19 year cyclical arrangement of the leap years beginning
from the first year of each pattern, the 19 year cycles appear to be

the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the 19 year
cycle

for the first selection and

the 3rd, 5th, 8th, 11th, 14th, 16th and 19th years of the 19 year cycle

for the second selection.

For some Hebrew year i let C(i) be a cycle of 19 consecutive
Hebrew year lengths beginning with the year length L(i).

The sequence of elements started at L(j) is the same as the sequence
of elements started at L(i-d). Unless d is a multiple of 19,
C(j) will appear to be different than C(i).

Since L(i) = L(i + 19*k) for any value of i and k,
19 apparent distributions exist for the common and leap years in a
mahzor katan.

It is common practice today to associate C(1) with a 19 year cycle
in which the order of leap years is the
3rd, 6th, 8th, 11th, 14th, 17th, and 19th year of the cycle. This
cycle is known by the acronym GUChADZaT, which is formed from
the Hebrew letters gimel-vov-het-alef-daled-zayin-tet. The
Hebrew letters also symbolize the digits 3, 6, 8, 1, 4, 7, and 9.

The 19 Leap Year Intercalation Cycles

The table below shows the 19 intercalation patterns which can be observed
relative to C(1).

The 19 Leap Year Intercalation Cycles

C(i)

Leap Year Order

Relative Order

Epochal Molad

C( 0)

1 4 7 9 12 15 18

2 3 3 2 3 3 3

3d 7h 695p

C( 1)

3 6 8 11 14 17 19

3 3 2 3 3 3 2

2d 5h 204p

C( 2)

2 5 7 10 13 16 18

3 3 2 3 3 3 2

6d 14h 0p

C( 3)

1 4 6 9 12 15 17

3 3 2 3 3 3 2

3d 22h 876p

C( 4)

3 5 8 11 14 16 19

3 2 3 3 3 2 3

2d 20h 385p

C( 5)

2 4 7 10 13 15 18

3 2 3 3 3 2 3

0d 5h 181p

C( 6)

1 3 6 9 12 14 17

3 2 3 3 3 2 3

4d 13h 1057p

C( 7)

2 5 8 11 13 16 19

2 3 3 3 2 3 3

3d 11h 566p

C( 8)

1 4 7 10 12 15 18

2 3 3 3 2 3 3

0d 20h 362p

C( 9)

3 6 9 11 14 17 19

3 3 3 2 3 3 2

6d 17h 951p

C(10)

2 5 8 10 13 16 18

3 3 3 2 3 3 2

4d 2h 747p

C(11)

1 4 7 9 12 15 17

3 3 3 2 3 3 2

1d 11h 543p

C(12)

3 6 8 11 14 16 19

3 3 2 3 3 2 3

0d 9h 52p

C(13)

2 5 7 10 13 15 18

3 3 2 3 3 2 3

4d 17h 928p

C(14)

1 4 6 9 12 14 17

3 3 2 3 3 2 3

2d 2h 724p

C(15)

3 5 8 11 13 16 19

3 2 3 3 2 3 3

1d 0h 233p

C(16)

2 4 7 10 12 15 18

3 2 3 3 2 3 3

5d 9h 29p

C(17)

1 3 6 9 11 14 17

3 2 3 3 2 3 3

2d 17h 905p

C(18)

2 5 8 10 13 16 19

2 3 3 2 3 3 3

1d 15h 414p

C(19)

1 4 7 9 12 15 18

2 3 3 2 3 3 3

6d 0h 210p

Note that C(19) = C(0).

The relative order cyclical pattern is formed from the number of
years that have elapsed since the previous leap year.

The relative order for C(4) is shown to be 3 2 3 3 3 2 3.
In Hebrew this numreic pattern is represented by letters
gimel-bet-tet-bet-gimel. Obviously, tet representing
9, reflects 3+3+3.

Since there are 7 leap years in a mahzor katan, only 7 relative order
patterns are produced.Hence, specifying a cycle by its relative order does
not uniquely identify the cycle. For example, it is to be noted that
gimel-bet-tet-bet-gimel is also a relative order for
C(5) and C(6).

The table also includes the Epochal Moladot which are
associated with the first 20 years of Yetsirah, that is,
Creation, or Genesis, or Aera Mundi.

Some References

1. Pages 91-95 of Studies In Hebrew Astronomy and Mathematics
by Solomon Gandz, KTAV Publishing House, Inc. (1970), show a study of
the calendar cycles as discussed in 3 medieval sources. These are the
Seder Olam, the Yesod Olam, and the
Pirke di Rabbi Eliezer (PdRE).

In this discussion, Gandz analyzes ancient rabbinic sources which
debate the issues of whether the mahzor katan ought to be either
C(1), C(2), C(3), C(4), or C(12). Gandz does not dicuss the fact
that cycles expressed in the relative order notation can refer to more
than one leap year intercalation cycle.

Gandz's text also indicates the existence of a third manner of noting the
leap year cycles. It seems to be the relative order notation, but with the
first entry indicating the number of years elapsed from the first year of
the cycle.

2. On page 48, under the subheading Historicalof the article
Calendar, in the Encyclopedia Judaica (1982),
the cycles C(1), C(2), C(3) and C(4) are shown together with their
Hebrew acronyms. The article identifies the epochal moladot associated with
C(1), C(2), and C(3) to be 2d 5h 204p, 6d 14h, and 3d 22h 876p respectively.

The article indicates that 4d 20h 408p
(named daled-hof-tof-het) is also an epochal molad.
This is a puzzling detail, since 4d 20h 408p represents
the molad of Heshvan 0H. Since the article shows C(4),
the value 2d 20h 385p would have been expected, since it represents
the value of the molad of Tishrei 4H.

3. Pages 64-66 of Al-Biruni's The Chronology of Ancient Nations
(1000g) as translated into English by Dr. E.C. Sachau in 1879g, indicate that
only 3 leap year cycles were in use at about 1000g (4761H). These cycles
were C(2) and C(3) "used by the Jews of Syria", and
C(4) "most extensively diffused among the
Jews; ... because they attribute its invention to Babylonia."

Although the cycle C(1) appears in documents that predate Al-Biruni's
work by several centuries, it is not mentioned at all
in the Chronology.

4. Chapters 6-8 of Maimonides' Hilchot Kiddush HaChodesh (1175g)
describe the calculation of the Hebrew calendar using C(1)
and BaHaRaD. Maimonides' instructions still remain in use today.

Calculating the Molad of Tishrei

Let M(i) represent the known value of the molad of Tishrei for some
year i.

Then M(i+1) = M(i) + L(i) is the molad of Tishrei for the
subsequent year i+1.

It is therefore not required to calculate LSUM(i,r) for more than
the first 18 terms.

The expression for LSUM(i,r) represents the sum of the first
r terms of C(i). Consequently, the evaluation
of LSUM(i,r) is dependent on the order of the
values of the length elements in C(i).

Before showing the tables for LSUM(i,r) these arithmetical
reductions need to be explained.

The moladot are normally expressed in units representing the day of the
week, the hour, and the number of halakim.

In traditional calculations, the moladot are remaindered using 1080
as the divisor. The quotient is then added to the hours. The hours are then
remaindered by 24 and the quotient added to the days. Finally,
the days are remaindered by 7. If the remainder for the day is
zero, it is sometimes expressed as 7 depending on the context.
Traditional texts show no entries for the halakim or the hours if their
corresponding value after remaindering is zero.

A 19 Hebrew year cycle (mahzor katan) consists of 235 months.
Since one month has the lunation period of 29d 12h 793p, 235 months
have

The following table shows the valuations for LSUM(i,r)
after the traditonal reductions.

The LSUM(i,r) Tables

LSUM( 0 , r)

LSUM( 1 , r)

LSUM( 2 , r)

LSUM( 3 , r)

1

5d 21h 589p

-

4d 8h 876p

-

4d 8h 876p

L

5d 21h 589p

-

2

3d 6h 385p

-

1d 17h 672p

L

3d 6h 385p

-

3d 6h 385p

-

3

0d 15h 181p

L

0d 15h 181p

-

0d 15h 181p

-

0d 15h 181p

L

4

6d 12h 770p

-

4d 23h 1057p

-

4d 23h 1057p

L

6d 12h 770p

-

5

3d 21h 566p

-

2d 8h 853p

L

3d 21h 566p

-

3d 21h 566p

L

6

1d 6h 362p

L

1d 6h 362p

-

1d 6h 362p

L

2d 19h 75p

-

7

0d 3h 951p

-

5d 15h 158p

L

0d 3h 951p

-

0d 3h 951p

-

8

4d 12h 747p

L

4d 12h 747p

-

4d 12h 747p

-

4d 12h 747p

L

9

3d 10h 256p

-

1d 21h 543p

-

1d 21h 543p

L

3d 10h 256p

-

10

0d 19h 52p

-

6d 6h 339p

L

0d 19h 52p

-

0d 19h 52p

-

11

5d 3h 928p

L

5d 3h 928p

-

5d 3h 928p

-

5d 3h 928p

L

12

4d 1h 437p

-

2d 12h 724p

-

2d 12h 724p

L

4d 1h 437p

-

13

1d 10h 233p

-

6d 21h 520p

L

1d 10h 233p

-

1d 10h 233p

-

14

5d 19h 29p

L

5d 19h 29p

-

5d 19h 29p

-

5d 19h 29p

L

15

4d 16h 618p

-

3d 3h 905p

-

3d 3h 905p

L

4d 16h 618p

-

16

2d 1h 414p

-

0d 12h 701p

L

2d 1h 414p

-

2d 1h 414p

L

17

6d 10h 210p

L

6d 10h 210p

-

6d 10h 210p

L

0d 22h 1003p

-

18

5d 7h 799p

-

3d 19h 6p

L

5d 7h 799p

-

5d 7h 799p

-

19

2d 16h 595p

L

2d 16h 595p

-

2d 16h 595p

-

2d 16h 595p

L

LSUM( 4 , r)

LSUM( 5 , r)

LSUM( 6 , r)

LSUM( 7 , r)

1

4d 8h 876p

-

4d 8h 876p

L

5d 21h 589p

-

4d 8h 876p

L

2

1d 17h 672p

L

3d 6h 385p

-

3d 6h 385p

L

3d 6h 385p

-

3

0d 15h 181p

-

0d 15h 181p

L

2d 3h 974p

-

0d 15h 181p

-

4

4d 23h 1057p

L

6d 12h 770p

-

6d 12h 770p

-

4d 23h 1057p

L

5

3d 21h 566p

-

3d 21h 566p

-

3d 21h 566p

L

3d 21h 566p

-

6

1d 6h 362p

-

1d 6h 362p

L

2d 19h 75p

-

1d 6h 362p

-

7

5d 15h 158p

L

0d 3h 951p

-

0d 3h 951p

-

5d 15h 158p

L

8

4d 12h 747p

-

4d 12h 747p

-

4d 12h 747p

L

4d 12h 747p

-

9

1d 21h 543p

-

1d 21h 543p

L

3d 10h 256p

-

1d 21h 543p

-

10

6d 6h 339p

L

0d 19h 52p

-

0d 19h 52p

-

6d 6h 339p

L

11

5d 3h 928p

-

5d 3h 928p

-

5d 3h 928p

L

5d 3h 928p

-

12

2d 12h 724p

-

2d 12h 724p

L

4d 1h 437p

-

2d 12h 724p

L

13

6d 21h 520p

L

1d 10h 233p

-

1d 10h 233p

L

1d 10h 233p

-

14

5d 19h 29p

-

5d 19h 29p

L

0d 7h 822p

-

5d 19h 29p

-

15

3d 3h 905p

L

4d 16h 618p

-

4d 16h 618p

-

3d 3h 905p

L

16

2d 1h 414p

-

2d 1h 414p

-

2d 1h 414p

L

2d 1h 414p

-

17

6d 10h 210p

-

6d 10h 210p

L

0d 22h 1003p

-

6d 10h 210p

-

18

3d 19h 6p

L

5d 7h 799p

-

5d 7h 799p

-

3d 19h 6p

L

19

2d 16h 595p

-

2d 16h 595p

-

2d 16h 595p

L

2d 16h 595p

-

LSUM( 8 , r)

LSUM( 9 , r)

LSUM(10 , r)

LSUM(11 , r)

1

5d 21h 589p

-

4d 8h 876p

-

4d 8h 876p

L

5d 21h 589p

-

2

3d 6h 385p

-

1d 17h 672p

L

3d 6h 385p

-

3d 6h 385p

-

3

0d 15h 181p

L

0d 15h 181p

-

0d 15h 181p

-

0d 15h 181p

L

4

6d 12h 770p

-

4d 23h 1057p

-

4d 23h 1057p

L

6d 12h 770p

-

5

3d 21h 566p

-

2d 8h 853p

L

3d 21h 566p

-

3d 21h 566p

-

6

1d 6h 362p

L

1d 6h 362p

-

1d 6h 362p

-

1d 6h 362p

L

7

0d 3h 951p

-

5d 15h 158p

-

5d 15h 158p

L

0d 3h 951p

-

8

4d 12h 747p

-

2d 23h 1034p

L

4d 12h 747p

-

4d 12h 747p

L

9

1d 21h 543p

L

1d 21h 543p

-

1d 21h 543p

L

3d 10h 256p

-

10

0d 19h 52p

-

6d 6h 339p

L

0d 19h 52p

-

0d 19h 52p

-

11

5d 3h 928p

L

5d 3h 928p

-

5d 3h 928p

-

5d 3h 928p

L

12

4d 1h 437p

-

2d 12h 724p

-

2d 12h 724p

L

4d 1h 437p

-

13

1d 10h 233p

-

6d 21h 520p

L

1d 10h 233p

-

1d 10h 233p

-

14

5d 19h 29p

L

5d 19h 29p

-

5d 19h 29p

-

5d 19h 29p

L

15

4d 16h 618p

-

3d 3h 905p

-

3d 3h 905p

L

4d 16h 618p

-

16

2d 1h 414p

-

0d 12h 701p

L

2d 1h 414p

-

2d 1h 414p

L

17

6d 10h 210p

L

6d 10h 210p

-

6d 10h 210p

L

0d 22h 1003p

-

18

5d 7h 799p

-

3d 19h 6p

L

5d 7h 799p

-

5d 7h 799p

-

19

2d 16h 595p

L

2d 16h 595p

-

2d 16h 595p

-

2d 16h 595p

L

LSUM(12 , r)

LSUM(13 , r)

LSUM(14 , r)

LSUM(15 , r)

1

4d 8h 876p

-

4d 8h 876p

L

5d 21h 589p

-

4d 8h 876p

-

2

1d 17h 672p

L

3d 6h 385p

-

3d 6h 385p

-

1d 17h 672p

L

3

0d 15h 181p

-

0d 15h 181p

-

0d 15h 181p

L

0d 15h 181p

-

4

4d 23h 1057p

-

4d 23h 1057p

L

6d 12h 770p

-

4d 23h 1057p

L

5

2d 8h 853p

L

3d 21h 566p

-

3d 21h 566p

L

3d 21h 566p

-

6

1d 6h 362p

-

1d 6h 362p

L

2d 19h 75p

-

1d 6h 362p

-

7

5d 15h 158p

L

0d 3h 951p

-

0d 3h 951p

-

5d 15h 158p

L

8

4d 12h 747p

-

4d 12h 747p

-

4d 12h 747p

L

4d 12h 747p

-

9

1d 21h 543p

-

1d 21h 543p

L

3d 10h 256p

-

1d 21h 543p

-

10

6d 6h 339p

L

0d 19h 52p

-

0d 19h 52p

-

6d 6h 339p

L

11

5d 3h 928p

-

5d 3h 928p

-

5d 3h 928p

L

5d 3h 928p

-

12

2d 12h 724p

-

2d 12h 724p

L

4d 1h 437p

-

2d 12h 724p

L

13

6d 21h 520p

L

1d 10h 233p

-

1d 10h 233p

L

1d 10h 233p

-

14

5d 19h 29p

-

5d 19h 29p

L

0d 7h 822p

-

5d 19h 29p

-

15

3d 3h 905p

L

4d 16h 618p

-

4d 16h 618p

-

3d 3h 905p

L

16

2d 1h 414p

-

2d 1h 414p

-

2d 1h 414p

L

2d 1h 414p

-

17

6d 10h 210p

-

6d 10h 210p

L

0d 22h 1003p

-

6d 10h 210p

-

18

3d 19h 6p

L

5d 7h 799p

-

5d 7h 799p

-

3d 19h 6p

L

19

2d 16h 595p

-

2d 16h 595p

-

2d 16h 595p

L

2d 16h 595p

-

LSUM(16 , r)

LSUM(17 , r)

LSUM(18 , r)

LSUM(19 , r)

1

4d 8h 876p

L

5d 21h 589p

-

4d 8h 876p

L

5d 21h 589p

-

2

3d 6h 385p

-

3d 6h 385p

L

3d 6h 385p

-

3d 6h 385p

-

3

0d 15h 181p

L

2d 3h 974p

-

0d 15h 181p

-

0d 15h 181p

L

4

6d 12h 770p

-

6d 12h 770p

-

4d 23h 1057p

L

6d 12h 770p

-

5

3d 21h 566p

-

3d 21h 566p

L

3d 21h 566p

-

3d 21h 566p

-

6

1d 6h 362p

L

2d 19h 75p

-

1d 6h 362p

-

1d 6h 362p

L

7

0d 3h 951p

-

0d 3h 951p

-

5d 15h 158p

L

0d 3h 951p

-

8

4d 12h 747p

-

4d 12h 747p

L

4d 12h 747p

-

4d 12h 747p

L

9

1d 21h 543p

L

3d 10h 256p

-

1d 21h 543p

L

3d 10h 256p

-

10

0d 19h 52p

-

0d 19h 52p

L

0d 19h 52p

-

0d 19h 52p

-

11

5d 3h 928p

L

6d 16h 641p

-

5d 3h 928p

-

5d 3h 928p

L

12

4d 1h 437p

-

4d 1h 437p

-

2d 12h 724p

L

4d 1h 437p

-

13

1d 10h 233p

-

1d 10h 233p

L

1d 10h 233p

-

1d 10h 233p

-

14

5d 19h 29p

L

0d 7h 822p

-

5d 19h 29p

-

5d 19h 29p

L

15

4d 16h 618p

-

4d 16h 618p

-

3d 3h 905p

L

4d 16h 618p

-

16

2d 1h 414p

-

2d 1h 414p

L

2d 1h 414p

-

2d 1h 414p

-

17

6d 10h 210p

L

0d 22h 1003p

-

6d 10h 210p

-

6d 10h 210p

L

18

5d 7h 799p

-

5d 7h 799p

-

3d 19h 6p

L

5d 7h 799p

-

19

2d 16h 595p

-

2d 16h 595p

L

2d 16h 595p

-

2d 16h 595p

L

Note that LSUM(0,r) is exactly the same as LSUM(19,r).

In the LSUM(i,r) tables, the leap years are indicated by a
capital L immediately to the right of the molad value.

Some Useful Relationships

The following relationships have a particular usefulness in the analysis
of Hebrew calendar calculation methods that do not imitate the instructions
found in chapters 6-8 of Maimonides' Hilchot Kiddush HaHodesh.

Note that since LSUM(7 , 17) shows no leap year indicator,
the year 5800H is a common year.

Relationship 6: LSUM(i,19*j) = (6939d 16h 595p)*j

Since L(i) = L(i+19*k) for any i and k, every sequence
of 19 consecutive Hebrew years has the mean lunar length
6939d 16h 595p.

Therefore, LSUM(i,19*j) = (6939d 16h 595p)*j

Modulo algebra shows that if a is congruent to
b modulo X, then a*k is congruent tob*k modulo X.

If the calculations are to produce only a molad value, rather than the
number of days in the calculated interval, then the 19 year excess
value of 2d 16h 595p can be substituted since it is congruent
to (6939d 16h 595p) modulo 7d.

Maimonides' Hilchot Kiddush HaHodesh, at 6:13, indicates that
by adding 2d 16h 595p to the molad at the beginning of a
19 year cycle, the molad at the beginning of the next 19 year cycle is
determined.

Relationship 7 shows that the addition of this value to any molad
of Tishrei will determine the value of the molad of Tishrei exactly 19
years away.

Al-Biruni's The Chronology of Ancient Nations, in pages 144-147
of Sachau's English translation, shows a method for calculating the molad
of Tishrei for any year x of the Aera Alexandri using
M(3461) and LSUM(3,r), so as to produce
a set of Tishrei moladot that demonstrate compliance to the cycle
C(4), whose Hebrew letter acronym is stated to be
gimel-bet-tet-bet-gimel.

Since 3461H represents the 12th year of the Aera Alexandri,
then according to Al-Biruni's method it was necessary that

M(x) = M(3461+x-12) = M(3461) + LSUM(3,x-12)

Relationship 7 helped to confirm Al-Biruni's method of dividing
x-12 by 19 to obtain both a multiplier m for
2d 16h 595p and a remainder r for use by LSUM(3,r).
Relationship 7 was also very helpful in identifying the many
descriptive errors in that part of the text.

Relationship 8: M(i) + LSUM(i,k) = M(k) + LSUM(k,i)

Since, M(i+k) = M(k+i)
M(i) + LSUM(i,k) = M(k) + LSUM(k,i) [Rel'n 3]

Example 8.0:Relationship 8 can be demonstrated using M(5759)
and the fact that M(5759) = M(1+5758) = M(5758+1).

Relationship 10 shows how to use any known M(i) together with
any arbitrarily chosen LSUM(j,r). Since M(i+k) can be
calculated using an LSUM(j,r) that may be different than
LSUM(i,r), LSUM(i, r) need not be developed for more than
one value of i.

Relationship 10 does not prove that one and only one intercalation
pattern exists for the Hebrew calendar since Relationship 10 is
derived from that assumption.

Example 10.0: Given the molad of Tishrei 5760H
relative to C(1) [GUChADZaT] how is the molad of
Tishrei 5775H determined using the table for LSUM(0,r)?

Relationship 11 shows that no matter how many multiples
of 19 are added to i-j
in Relationship 10 the result will
always be the same since the multiples cancel each other in the expression.
Consequently, whenever i-j < 0 as many multiples of 19 may be added
to i-j so as to make 19*m+i-j a positive (or zero) value.

Example 11.0: Given the molad of Tishrei 5760H
relative to C(1) [GUChADZaT] how is the molad of Tishrei 5775H
determined using the table corresponding to LSUM(5767,r)?

LSUM(1,12) and LSUM(1,17) can be found in the
LSUM(1,r) table, thereby bypassing the use of Shocken's formula.
However, bypassing the formula causes a loss of the day count between
M(5770) and M(5775).

Conclusion

In light of today's electronic computers, the LSUM(i,r) tables are of
limited use. However, the logical principles which establish their data
will be required for quite some time into the future and continue to be
useful in analysing the ancient Hebrew calendar.

The analysis produced an unexpected result.

The relationship

M(i+k) = M(i) + L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)

indicates that the Hebrew calendar is entirely constant
once the equation is given these three initial conditions:-

1. some value for year i
2. some value M(i) for its corresponding molad of Tishrei
3. some intercalation pattern C(i) beginning at year i

After setting these three initial conditions, the value of each and every
Tishrei molad M(i) is fixed over the entire life
of the Hebrew calendar.

It is not known today which combination of i, M(i), and C(i) was
originally chosen by the ancient scholars of the Hebrew calendar.
However, based on the foregoing analysis, it is very likely that their
original values exist in the current set of Tishrei moladot defined by

This combination of values did not appear in all of the ancient
Hebrew calendar documents. It eventually became a current standard
by at least the 12th century c.e., as evidenced from Maimonides' work
Hilchot Kiddush HaHodesh.

Hence, the analysis confirmed that the Hebrew calendar, set into motion so
many centuries ago, has endured steadfast against the intellectual storms of
its ages. In this manner, the Hebrew calendar has remained one of the
brightest beacons to the memory of the scholars who took part in its
creation.